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A180608 O.g.f.: exp( Sum_{n>=1} A067692(n)*x^n/n ), where A067692(n) = [sigma(n)^2 + sigma(n,2)]/2.

%I A180608 #15 Oct 29 2024 07:07:49
%S A180608 1,1,4,8,21,39,93,171,364,675,1338,2433,4641,8282,15222,26811,47920,
%T A180608 83046,145288,248164,425970,718303,1213106,2020540,3365352,5541996,
%U A180608 9115640,14856657,24164430,39002462,62800603,100454208,160257140
%N A180608 O.g.f.: exp( Sum_{n>=1} A067692(n)*x^n/n ), where A067692(n) = [sigma(n)^2 + sigma(n,2)]/2.
%C A180608 sigma(n) = A000203(n), sum of divisors of n;
%C A180608 sigma(n,2) = A001157(n), sum of squares of divisors of n.
%H A180608 Vaclav Kotesovec, <a href="/A180608/b180608.txt">Table of n, a(n) for n = 0..10000</a>
%F A180608 log(a(n)) ~ 3*(7*zeta(3))^(1/3) * n^(2/3) / 2^(4/3). - _Vaclav Kotesovec_, Oct 29 2024
%e A180608 O.g.f.: A(x) = 1 + x + 4*x^2 + 8*x^3 + 21*x^4 + 39*x^5 + 93*x^6 +...
%e A180608 log(A(x)) = x + 7*x^2/2 + 13*x^3/3 + 35*x^4/4 + 31*x^5/5 + 97*x^6/6 +...
%t A180608 nmax = 40; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[(DivisorSigma[1, k]^2 + DivisorSigma[2, k])/2*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* _Vaclav Kotesovec_, Oct 28 2024 *)
%o A180608 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, (sigma(m)^2+sigma(m,2))/2*x^m/m)+x*O(x^n)), n)}
%Y A180608 Cf. A067692, A000203, A001157, A072861, A156302.
%K A180608 nonn
%O A180608 0,3
%A A180608 _Paul D. Hanna_, Oct 10 2010